A013618 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(45 + 99\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2025 + 8910\,x + 9801\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	2025	8910	9801															20736
1	91125	601425	1323135	970299														2985984
2	4100625	36085500	119082150	174653820	96059601													429981696
3	184528125	2029809375	8931161250	19648554750	21613410225	9509900499												61917364224
4	8303765625	109609706250	602853384375	1768369927500	2917810380375	2567673134730	941480149401											8916100448256
5	373669453125	5754509578125	37979763215625	139259131790625	306370089939375	404408518719975	296566247061315	93206534790699										1283918464548864
6	16815125390625	295946206875000	2278785792937500	10026657488925000	27573308094543750	48529022246397000	53381924471036700	33554352524651640	9227446944279201									184884258895036416
7	756680642578125	14982276723046875	131844035162812500	676799380502437500	2233437955658043750	4913563502447696250	7206559803589954500	6794756386241957100	3737116012433076405	913517247483640899								26623333280885243904
8	34050628916015625	749113836152343750	7416226977908203125	43508531603728125000	167507846674353281250	442220715220292662500	810737977903869881250	1019213457936293565000	840851102797442191125	411082761367638404550	90438207500880449001							3833759992447475122176
9	1532278301220703125	37081134889541015625	407892483784951171875	2692090392980677734375	11845197729114982031250	36483209005674144656250	80263059812483118243750	126127665419616328668750	138740431961577961535625	101742983438490505126125	44766912712935822255495	8953382542587164451099						552061438912436417593344
10	68952523554931640625	1820346621850195312500	22026194124387363281250	161525423578840664062500	799550846715261287109375	2814418980437719730625000	7223675383123480641937500	13621787865318563496225000	18729958314813024807309375	18313737018928290922702500	12087066432492672008983650	4834826572997068803593460	886384871716129280658801					79496847203390844133441536
11	3102863559971923828125	88741897815197021484375	1171393051160600683593750	9449237279362178847656250	51970805036491983662109375	205804387944508255301953125	603692871303890882219062500	1328124316868559940881937500	2191405122833123902455196875	2678384039018262547445240625	2356977954336071041751811750	1414186772601642625051087050	518535149953935629185398585	87752102299896798785221299				11447545997288281555215581184
12	139628860198736572265625	4300568894121086425781250	61498135185931535888671875	541183589636197515820312500	3274160717298994970712890625	14406307156115577871136718750	47540813615181406974751171875	119531188518170394679374375000	230097537897478009757795671875	337476388916301080978100318750	371224027807931189075910350625	296979222246344951260728280500	163338572235489723193400554275	55283824448934983234689418370	8687458127689783079736908601			1648446623609512543951043690496
13	6283298708943145751953125	207348857395123809814453125	3193172403884906671142578125	30441576917036110264892578125	200914407652438327748291015625	972425733037801506301728515625	3565561021138605523106337890625	10085444031220627051072212890625	22187976868685379512358868359375	37966093753083871610036285859375	50115243754070710525247897334375	50115243754070710525247897334375	36751178752985187718515124711875	18658290751515556841707678699875	5864034236190603578822413305675	860058354641288524893953951499		237376313799769806328950291431424
14	282748441902441558837890625	9952745154965942871093750000	164220295056938057373046875000	1685995029251230722363281250000	12054864459146299664897460937500	63649684344292462230658593750000	256720393521979597663656328125000	806835522497650164085777031250000	1996917918181684156112298152343750	3905083928888626794175160831250000	6013829250488485263029747680125000	7216595100586182315635697216150000	6615212175537333789332722448137500	4477989780363733642009842887970000	2111052325028617288376068790043000	619242015341727737923646845079280	85145777109487563964501441198401	34182189187166852111368841966125056

Technical details

Node id: n-436c06bbad4746ea · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-2235b7dd30e0ad6a

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:                2025,                 8910,                  9801;
  1:               91125,               601425,               1323135,                 970299;
  2:             4100625,             36085500,             119082150,              174653820,                96059601;
  3:           184528125,           2029809375,            8931161250,            19648554750,             21613410225,              9509900499;
  4:          8303765625,         109609706250,          602853384375,          1768369927500,           2917810380375,           2567673134730,            941480149401;
  5:        373669453125,        5754509578125,        37979763215625,        139259131790625,         306370089939375,         404408518719975,         296566247061315,           93206534790699;
  6:      16815125390625,      295946206875000,      2278785792937500,      10026657488925000,       27573308094543750,       48529022246397000,       53381924471036700,        33554352524651640,         9227446944279201;
  7:     756680642578125,    14982276723046875,    131844035162812500,     676799380502437500,     2233437955658043750,     4913563502447696250,     7206559803589954500,      6794756386241957100,      3737116012433076405,       913517247483640899;
  8:   34050628916015625,   749113836152343750,   7416226977908203125,   43508531603728125000,   167507846674353281250,   442220715220292662500,   810737977903869881250,   1019213457936293565000,    840851102797442191125,    411082761367638404550,    90438207500880449001;
  9: 1532278301220703125, 37081134889541015625, 407892483784951171875, 2692090392980677734375, 11845197729114982031250, 36483209005674144656250, 80263059812483118243750, 126127665419616328668750, 138740431961577961535625, 101742983438490505126125, 44766912712935822255495, 8953382542587164451099;

%F Formula

The row polynomials satisfy P(n,x) = (45+99*x)*P(n-1,x) for n >= 1, with P(0,x) = 2025 + 8910*x + 9801*x^2.

%t Mathematica

P[0, x_] := 2025 + 8910*x + 9801*x^2;
P[n_, x_] := P[n, x] = Expand[(45 + 99*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2025 + 8910*x + 9801*x^2; for(n = 1, N, my(i = n + 1); v[i] = (45 + 99*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle of coefficients in expansion of (1+11x)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (1+11x)^n.?
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.